| L(s) = 1 | + (14.7 − 17.6i)2-s + (56.3 − 236. i)3-s + (85.8 + 487. i)4-s + (1.65e3 − 4.54e3i)5-s + (−3.33e3 − 4.48e3i)6-s + (846. − 4.80e3i)7-s + (3.02e4 + 1.74e4i)8-s + (−5.26e4 − 2.66e4i)9-s + (−5.56e4 − 9.63e4i)10-s + (−3.64e3 − 1.00e4i)11-s + (1.19e5 + 7.15e3i)12-s + (4.18e5 − 3.51e5i)13-s + (−7.21e4 − 8.59e4i)14-s + (−9.80e5 − 6.46e5i)15-s + (2.79e5 − 1.01e5i)16-s + (−2.31e6 + 1.33e6i)17-s + ⋯ |
| L(s) = 1 | + (0.462 − 0.550i)2-s + (0.231 − 0.972i)3-s + (0.0838 + 0.475i)4-s + (0.529 − 1.45i)5-s + (−0.428 − 0.577i)6-s + (0.0503 − 0.285i)7-s + (0.923 + 0.533i)8-s + (−0.892 − 0.451i)9-s + (−0.556 − 0.963i)10-s + (−0.0226 − 0.0622i)11-s + (0.482 + 0.0287i)12-s + (1.12 − 0.945i)13-s + (−0.134 − 0.159i)14-s + (−1.29 − 0.851i)15-s + (0.266 − 0.0970i)16-s + (−1.63 + 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.860867 - 2.61598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860867 - 2.61598i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-56.3 + 236. i)T \) |
| good | 2 | \( 1 + (-14.7 + 17.6i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (-1.65e3 + 4.54e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (-846. + 4.80e3i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (3.64e3 + 1.00e4i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-4.18e5 + 3.51e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (2.31e6 - 1.33e6i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-5.49e5 + 9.51e5i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (6.29e6 - 1.11e6i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (-1.16e7 + 1.38e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (2.63e6 + 1.49e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (1.55e7 + 2.69e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-1.33e8 - 1.59e8i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-2.26e8 + 8.23e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-1.85e8 - 3.27e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 4.06e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.56e8 + 7.03e8i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (1.57e8 - 8.95e8i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-4.17e8 + 3.50e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (-7.18e8 + 4.14e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-7.90e8 + 1.36e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.58e9 + 1.33e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (1.98e9 - 2.36e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-1.32e9 - 7.66e8i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-6.49e9 + 2.36e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.70645941452705729298727658579, −13.16120586114414847860101937647, −12.40587484939301049463952395349, −11.02915570456194457722822385078, −8.843925458241782750083186029542, −7.932943572395645749054458153259, −5.94591558728073260619332034260, −4.14406187705689153515096033635, −2.22392669577146636020360485087, −0.908436105859014161493447521507,
2.34395112176892841125388019320, 4.14377136189425688855044795103, 5.81192283653936286737165940418, 6.88588908913295239683454296337, 9.115010079230242717752700305342, 10.43018956933171523243902938272, 11.20310289643419452334576336268, 13.94008912502002453542650108804, 14.15002636319159652197191189677, 15.49004351065328913497296868740
